Elliptic Systems with Nonlinearities of Arbitrary Growth

Abstract.

In this paper we study the existence of nontrivial solutions for the following system of coupled semilinear Poisson equations: $\left\{ {\begin{array}{*{20}l} { - \Delta u = v^p ,} & {{\text{in }}\Omega ,} \\ { - \Delta v = f(u),} & {{\text{in }}\Omega ,} \\ {u = 0{\text{ and }}v = 0,} & {{\text{on }}\partial \Omega ,} \\ \end{array} } \right.$ where Ω is a bounded domain in $\mathbb{R}^N .$ We assume that $0 < p < \frac{2} {{N - 2}},$ and the function f is superlinear and with no growth restriction (for example f(s) = s e^s); then the system has a nontrivial (strong) solution.